Linear Inequalities

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In mathematics, linear inequality is an inequality that involves a linear function. Linear inequality has one of the symbols of inequality. Shows data that is not equal in the form of a graph.


Symbol of Inequality:

  • < less than

  • > greater than

  • ≤ less than or equal to

  • ≥ greater than or equal to

  • ≠ not equal to

  • = equal to

Linear Inequation of One Variable

Let a be the non-zero real number and x be a variable. In one variable, inequalities of the form ax + b > 0, ax + b < 0, ax + b?? 0 and ax + b < 0 are known as linear inequalities.


Linear Inequation of Two Variables

Let x, y be variables, and a, b be non-zero real numbers. The inequation of the form ax + by < c, ax + by > c, ax + by ≤ c, and ax + by ≥ c are then referred to as linear inequities in the two variables x and y.


Rules for Solving Inequalities

There are similar rules for solving inequalities to those for solving linear equations. However, when multiplying or dividing by a negative number, there is one exception. In order to solve inequalities, we can:

  • Add the same number on both sides. 

  • From both sides, subtract the same number. 

  • By the same positive number, multiply both sides. 

  • By the same positive number, divide both sides. 

  • Multiply the same negative number on both sides and reverse the sign. 

  • Divide the same negative number between both sides and reverse the sign.

Ex: Solve x – 6 > 14

Solution:

x – 6 > 14

x – 6+ 6 > 14 + 6

x > 20


Ex. Solve the inequality 12 > 18 – y

Solution:

12 > 18 – y

18 – y < 12

18 – y – 18 < 12 –18

– y < –6

y > 6

How to Represent the Solution of a Linear Inequality in One Variable on a Number Line

We use the following conventions to depict the solution of a linear inequality in one variable on a number line:

  • If the inequality involves ‘≥’ or ‘≤’, we draw a filled circle(•) on the number line to indicate that the number corresponding to the filled circle is included in the solution set. 

  • If the inequality involves ‘>’ or ‘<’, we draw an open circle (O) on the number line to indicate that the number corresponding to the open circle is excluded from the solution set. 

How to Graphically Represent the Solution of a Linear Inequality 

1. To represent the solution of a linear inequality in a plane graphically in one or two variables, we proceed as follows:

  1. If the inequality involves ‘≥’ or ‘≤’, we draw the graph of the line as a thick line to indicate that the points on this line are included in the solution set. 

  2. If the inequality involves ‘>’ or ‘<’, we draw the graph of the line as a dotted line to indicate that the points on the line are excluded from the solution set.

2. Solution of a linear inequality in one variable can be represented on the number line as well as in the plane but the solution of a linear inequality in two variables of the type ax + by > c, ax + by ≥ c, ax + by < c or ax + by ≤ c (a ≠ 0, b ≠ 0) can be represented in the plane only.

3. A system of inequalities contains two or more inequalities taken together, and the solutions to the system of inequalities are the solutions common to all the inequalities that form the system.


Important Results

  1. If a, b ∈ R and b ≠ 0, then 

(i) ab > 0 or a b > 0 ⇒ a and b are of the same sign. 

(ii) ab < 0 or a b < 0 ⇒ a and b are of opposite sign. 

  1. If a is any positive real number, i.e., a > 0, then 

(i) | x | < a ⇔ – a < x < a 

     | x | ≤ a ⇔ – a ≤ x ≤ a 

(ii) | x | > a ⇔ x < – a or x > a 

     | x | ≥ a ⇔ x ≤ – a or x ≥ a

FAQs (Frequently Asked Questions)

1. How Can One Solve Linear Inequalities?

Answer: Linear inequalities can be solved through: 

  1. The equation is rearranged in such a way that "y" is on the left, and everything else is on the right. 

  2. Plotting of the “y=” line. For y< or y>  make a dashed line and for y≤ or y≥ make a solid line.

  3. In the case of above the line, one must shade for a “greater than” (y> or y≥) or below the line for a “less than” (y< or y≤).

2. What Symbols are Used in Linear Inequalities?

Answer: List of the symbol of inequality:

  • < less than

  • > greater than

  • ≤ less than or equal to

  • ≥ greater than or equal to

  • ≠ not equal to

  • = equal to

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